A Markov Approach to the Average Run Length of Cusum Charts for an AR(1) Process
Some papers have presented the effects of autocorrelated observations to Cusum charts. In this paper, the performance of Cusum charts in a first-order autoregressive (AR(1)) model is examined in some more detail by representing the Cusum statistic by a two-dimensional Markov process. The distribution of Cusum statistic at any time and the Average Run Length (ARL) can be analytically derived. Thus a conditional out of control ARL for any change point can be obtained. The variation pattern of the out of control ARL can be presented until the ARL converges to the steady state ARL. It can be shown that the variation pattern in the case of the no autocorrelation is monotonously decreasing in the change point. In the presence of autocorrelation, the monotonious decrese still holds, however, only when the autocorrelation is not highly positive.
KeywordsChange Point Control Chart Cusum Chart Positive Autocorrelation Negative Autocorrelation
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